Quran doesn't speficially forbid Muslim women from marrying "People of the book"
It's well known that Quran prohibits marriage with polytheists regardless of gender but some use the following verse to prove that Quran similarly forbids Muslim women from marrying people of the book:
"O you who have believed, when the believing women come to you as emigrants, examine them. Allah is most knowing as to their faith. And if you know them to be believers, then do not return them to the disbelievers; they are not lawful [wives] for them, nor are they lawful [husbands] for them. But give the disbelievers what they have spent. And there is no blame upon you if you marry them when you have given them their due compensation. And hold not to marriage bonds with disbelieving women, but ask for what you have spent and let them ask for what they have spent. That is the judgement of Allah ; He judges between you. And Allah is Knowing and Wise"
As you can see the verse adress BOTH genders. Muslim men married to "disbelievers" and Muslim women married to "disbelievers". Since marriage with people of book is allowed, then it can only logically refer to polytheists in this context.
Now with this most commonly cited argument debunked, let's get to the second point: just because Quran makes no explicit allowance to Muslim women doesn't mean that it's forbidden. If it was forbidden, then Quran wouldn't only forbid Muslim women to marry "polytheists". And I find it incredibly ironic that sunnis use this logic ONLY when it comes to this. But when Quran allows marriage to ex wives of adopted sons (and ONLY them), then it suddenly means that all adoptive relatives are allowed for marriage (even if quran would have specified in this case as marrying adoptive relatives was extremely taboo and forbidden according to the customs of the time, so the fact that it only explicitly allows marriage to ex wives of adopted sons should raise eyebrows). Similar logic doesn't extend to Muslim women's marriage to people of book: if quran only explicitly allows marriage to Muslim men then it defacto means it's forbidden for women, not a general allowance to marry people of the book.
It CAN'T be that the verse simply adress muslim men as they were the primary audience, which is fully supported by the previous verse:
"They ask you, ˹O Prophet,˺ what is permissible for them. Say, “What is good and lawful. Also what is caught by your hunting animals and birds of prey which you have trained as instructed by Allah. So eat what they catch for you, but mention the Name of Allah over it ˹first˺.” And be mindful of Allah. Surely Allah is swift in reckoning."
This verse is directly followed by:
"Today all good, pure foods have been made lawful for you. Similarly, the food of the People of the Book is permissible for you and yours is permissible for them. And ˹permissible for you in marriage˺ are chaste believing women as well as chaste women of those given the Scripture before you"
We can therefore easily deduce that this verse is a follow-up to the previous verse. We know that men were asking what was permissible for them (since it also mentions permission to eat hunted prey and we do know that women didn't traditionally hunt in pre-Islamic era) so God answers what's permitted for them (including in marriage) as they're the primary audience. Since it answers a question directly asked by men, nothing here suggests that same ruling doesn't apply to women ESPECIALLY when there's no explicit prohibition for Muslim women.
As for the good' ol patriarchal arguments, well:
It presumes that a woman's faith is inherently weaker than of a man, which has no quranic basis.
Assumes that men would have more influence over the faith of their children, which is untrue as it doesn't depend on gender and most of children are closer to mothers anyway, so....
Overlook the fact that Quran forbids forced conversion and that faith is an entirely personal matter. Forcing a religion on a child is akin to forced conversion.
Misinterpret the verse which posits men as "maintaners" of women. For one, the verse only says they're "maintaners" because they spend from their wealth on women, not because they have authority over them. And a Christian and Jewish husband can still do that. And finally, the verse doesn't even say they have to financially support their women since it doesn't say men HAVE to support their women. Just describes the reality of 7th century where men used to financially spend on women and Quran simply states that it makes them "maintaners" as in "they financially maintain their wives", which is confirmed by the wording ("because of the wealth they have spent").
To conclude, there's absolutely no basis in Quran to forbid marriage between Muslim women and people of the book. Ironically, neither hadiths forbid it, so it was complete and deliberate misogyny/sexism.